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Ch_04 - Vector II - Exam_Soln

# Ch_04 - Vector II - Exam_Soln - Exam Problems of Vector...

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Exam Problems of Vector Integral Calculus

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Vector Integral Calculus Line Integrals Line Integrals 1. Line Integral Evaluate ' C ds Fr , 22 1 () [ , , 0 ] x yy x  F where C is the circle with 1 xy  , and 0 z , oriented clockwise. Solution: The curve is the circle with radius of 1, by letting cos x , sin y , 02 = 0 = c o s s i n   ri j k i j k It has the unit tangent vector '( ) = sin cos j The vector F is represented as parametric form as 1 [ , , 0 ] s i n c o s 0 y x Fi j k Then 2 0 ' [(sin )(sin ) (cos )(cos )] 2 C ds d   ※ 원점에서 F 불연속이므로 Stoke’s 정리를 사용할 없다 .
Vector Integral Calculus Line Integrals 2. (’08) Line Integral Calculate the line integral with respect to arc length 22 () C x yd s where C has the vector equation of ( ) (cos sin ) (sin cos ) ri j tat t t at tt  ( 02 t ) Solution: Ans: 23 2 2( 1 2 ) a Recall: The arc length of a curve C from point A to (, ,) xyz is given by  222 /// b a s dx dt dy dt dz dt dt ds dx dt dy dt dz dt dt where x xt , yy t , zz t is the parameterization of C . 2 2 (cos 2 cos sin xy a t t tt  2 sin sin 2 cos sin t t t t  cos ) (1 ) 2 2 ' ( ) ( sin sin cos ) (cos cos sin ) cos sin ' () (cos s in ) j ij r ta t t t t t at t at t ttt a t      Then, 2 3 3 0 ' ( ) ( ) r CC t s x y td t attd t   Calculate by parts for 2 33 0 t at td t : 2 24 2 3 3 2 4 2 3 2 0 0 ( ) 1 2 ) t t t a a a    

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Vector Integral Calculus Line Integrals 3. (’08) Line Integral Derive the Maxwell equations based on criterion for exactness and independence of path. (A) H UP V  : Enthalpy (B) A UT S  : Helmholtz energy (C) GH T S : Gibbs energy Solution: Maxwell equations We get dU TdS PdV dH dU PdV VdP TdS VdP dA dU TdS SdT PdV SdT dG dH TdS SdT VdP SdT For line integral, Criteria for exactness and independence of path  123 CC dF d x F d y F d z  Fr is independent of path 0 curl   FF : exactness or 21 F F x y  , 3 2 F F yz , 3 1 F F zx Now 12 dF F dx F dy with F F x y dU TdS PdV dF F dx F dy SV TP VS    dH TdS VdP dF F dx F dy SP TV PS dA PdV SdT dF F dx F dy VT dG VdP SdT dF F dx F dy PT
Vector Integral Calculus Line Integrals 4. (’09) Line Integral Calculate the work done by force field (, ,) ( ) ( ) ( ) xyz y z z x x y  Fi j k along the curve intersection of the sphere 222 4  and the plane tan zy , where 0 / 2  . The path is traversed in a direction that appears counterclockwise when viewed from high above the xy -plane.

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Ch_04 - Vector II - Exam_Soln - Exam Problems of Vector...

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